Mathematics for the interested outsider

Cartan’s Formula

It starts with the observation that for a function and a vector field , the Lie derivative is and the exterior derivative evaluated at is . That is, on functions.

Next we consider the differential of a function. If we apply to it, the nilpotency of the exterior derivative tells us that we automatically get zero. On the other hand, if we apply , we get , which it turns out is . To see this, we calculate

just as if we took and applied it to .

So on exact -forms, gives zero while gives . And on functions gives , while gives zero. In both cases we find that

and in fact this holds for all differential forms, which follows from these two base cases by a straightforward induction. This is Cartan’s formula, and it’s the natural extension to all differential forms of the basic identity on functions.

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.